If you have a new computer, it's most likely running Mtn Lion. Did you migrate from your old machine on first boot of your new one? If not, I suggest that you boot into recovery mode, wipe the HD, and then restore the OS. On first boot, use the Setup Assistant to migrate from either the old machine or from the TM backup, following the steps outlined in Pondini's Setup New Mac guide. That should mitigate all your issues.
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I have my old computer backed up on my external hard drive via Time Machine. I have a new computer now and would like to be able to browse the external hard drive for specific files from the old computer. However, all of my backups from the old one are locked and not accessible from my new computer. Is there a way around this?
Yes. See the A New Mac section of Time Machine - Frequently Asked Question #19, especially the last sentence there, in red.
From another Mac, yes, but there may be complications.
Use the (hidden) Browse Other Backup Disks option, per Time Machine - Frequently Asked Question #17.
The complication is, the user account on your work Mac may not be recognized as being the same one as on the backups. THe cause is different, but the pink box in Problems after using Migration Assistant has an explanation of why.
You may need to use a different account, or create one or more, to find one with the same UID as on the backups. Once you do, if you restore the needed files to that account, the other account still probably won't have permission to them.
So either restore them to an external disk that doesn't enforce permissions, or to the special /Users/Shared folder. Then log on to the account that wants the files and copy from there. For either of those options, do not use the normal Restore button at the lower right; instead, select the item(s), control-click, and choose the Restore <items> to...option and choose the destination.